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Hilbert system of axioms

A system of axioms proposed by D. Hilbert [1] in 1899, and subsequently amended and made more precise by him.

In Hilbert's system of axioms the primary (undefined) notions are points, straight lines, planes, and the relations between them are expressed by the words "belongs to" , "between in the Hilbert system of axiomsbetween" and "congruence in geometrycongruent to in the Hilbert system of axiomscongruent to" . The nature of the primary objects and the relations between those objects are arbitrary as long as the objects and the relations satisfy the axioms.

Hilbert's system contains 20 axioms, which are subdivided into five groups.

Contents

Group

. For any two points there exists a straight line passing through them.

. There exists only one straight line passing through any two distinct points.

. At least two points lie on any straight line. There exist at least three points not lying on the same straight line.

. There exists a plane passing through any three points not lying on the same straight line. At least one point lies on any given plane.

. There exists only one plane passing through any three points not lying on the same straight line.

. If two points and of a straight line lie in a plane , then all points of lie in .

. If two planes have one point in common, then they have at least one more point in common.

. There exist at least four points not lying in the same plane.

Group

comprises four order axioms describing the relation "between" .

. If a point lies between a point and a point , then , and are distinct points on the same straight line and also lies between and .

. For any two points and on the straight line there exists at least one point such that the point lies between and .

. Out of any three points on the same straight line there exists not more than one point lying between the other two.

(Pasch's axiom). Let , and be three points not lying on the same straight line, and let be a straight line in the plane not passing through any of the points , or . Then, if the straight line passes through an interior point of the segment , it also passes through an interior point of the segment or through an interior point of the segment .

Group

comprises five axioms of congruence, which describe the relation of "being congruent" (Hilbert denoted this relation by the symbol ).

. Given a segment and a ray , there exists a point on such that the segment is congruent to the segment , i.e. .

. If and , then .

. Let and be two segments on a straight line without common interior points, and let and be two segments on the same or on a different straight line, also without any common interior points. If and , then .

. Let there be given an angle , a ray and a half-plane bounded by the straight line . Then contains one and only one ray such that . Moreover, every angle is congruent to itself.

. If for two triangles and one has , , , then .

Group

comprises two continuity axioms.

(Archimedes' axiom). Let and be two arbitrary segments. Then the straight line contains a finite set of points such that the point lies between and , the point lies between and , etc., and such that the segments are congruent to the segment , and lies between and .

(Cantor's axiom). Let there be given, on any straight line , an infinite sequence of segments which satisfies two conditions: a) each segment in the sequence forms a part of the segment which precedes it; b) for each preassigned segment it is possible to find a natural number such that . Then contains a point belonging to all the segments of this sequence.

Group

comprises one axiom about parallels. Let there be given a straight line and a point not on that straight line. Then there exists not more than one straight line passing through , not intersecting and lying in the plane defined by and .

(Hilbert classified the axiom about parallels in Group IV, and the continuity axioms in Group V).

All other axioms of Euclidean geometry are defined by the basic concepts of Hilbert's system of axioms, while all the statements regarding the properties of geometrical figures and not included in Hilbert's system must be logically deducible from the axioms, or from statements which are deducible from these axioms.

Hilbert's system of axioms is complete; it is consistent if the arithmetic of real numbers is consistent. If, in Hilbert's system, the axiom about parallels is replaced by its negation, the new system of axioms thus obtained is also consistent (the system of axioms of Lobachevskii geometry), which means that the axiom about parallels is independent of the other axioms in Hilbert's system. It is also possible to demonstrate that some other axioms of this system are independent of the others.